Modeling Non Newtonian Fluid Invasion Into Reservoir Rocks
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Proceedings of COBEM 2005 20th International Congress of Mechanical Engineering Copyright © 2009 by ABCM November 15-20, 2009, Gramado, RS, Brazil MODELING NON NEWTONIAN FLUID INVASION INTO RESERVOIR ROCKS Alex Tadeu de Almeida Waldmann PETROBRAS- Cidade Universitária, Q7 – Ilha do Fundão – Prédio 20 – Sala 1017 – RJ - CEP: 21941-598 [email protected] Cristiano Dannenhauer ESSS – Rua Lauro Müller, 116 – Torre do Rio Sul – 14º Andar, sala 1404 - Botafogo [email protected] Alex Rodrigues de Andrade BAKER HUGHES - Rua Maria Francisca Borges Rêgo Rei, 363, Macaé - RJ, CEP: 27933-260 [email protected] Idvard Pires Jr. BAKER HUGHES - Rua Maria Francisca Borges Rêgo Rei, 363, Macaé - RJ, CEP: 27933-260 [email protected] Andre Leibsohn Martins PETROBRAS- Cidade Universitária, Q7 – Ilha do Fundão – Prédio 20 – Sala 1017 – RJ - CEP: 21941-598 [email protected] Abstract. Minimizing fluid invasion is a major issue while drilling reservoir rocks. Large invasion may create several problems in sampling reservoir fluids in exploratory wells. Unreliable sampling may lead to wrong reservoir evaluation and, in critical cases, to wrong decisions concerning reservoir exploitability. Besides, drilling fluid invasion may also provoke irreversible reservoir damage, reducing its initial and /or its long term productivity (Ladva et al., 2000). Such problem can be critical in heavy oil reservoirs, where oil and filtrate interaction can generate stable emulsions. Invasion in light oil reservoir is less critical due to its good mobility properties. Other critical scenario is the low permeability gas reservoirs where imbibition effects may result in deep invasion. A common practice in the industry is the addition of bridging agents, such as calcium carbonates in the drilling fluid composition. Such products would form a low permeability layer at the well walls which would control invasion. An adequate drilling fluid design requires bridging agent size distribution and concentration optimization. The ability of the fluid system to prevent invasion is normally evaluated by standardized static filtration experiments. In these tests, the fluid is pressurized through a filter paper or into a consolidated inert porous medium. The volume which crosses the porous core is monitored along the time. This article presents the Darcy flow modeling of non-compressible cakes to reproduce adequately the filtration of a Non Newtonian fluid + particulate system through porous medium and non Newtonian radial flow modeling to predict the invasion profile into reservoir. Pressure differential, rheological behavior, filter cake permeability proved to be relevant parameters affecting the invasion profile. Keywords : Drilling fluid invasion, Non-Newtonian radial flow modeling, Reservoir rocks 1. INTRODUCTION In petroleum engineering, well drilling is an area of continuous development in order to improve the current technologies and look for new ones which can be applied to the adverse conditions faced nowadays and enable operations that were only conceptual some decades ago. Oil well drilling involves costly operations where avoiding reservoir damage and minimizing operational time are very important issues. The drilling operation generally occurs through weight and rotation of a string which extremity is connected to a bit. Simultaneously, a drilling fluid is circulated through the well according to the following path: the fluid is injected into the column, passes through the bit nozzles and returns through the annular space formed by the wellbore and drilling column. Figure 1 highlights the fluid circulation scheme. Figure 1 - Fluid circulation system Different types of fluid are used in the several the drilling phases of an offshore well. In its initial phases, the well is drilled without fluid return, with sea water or water with clay when higher densities are required. Extended and high angle sections are normally drilled with fluids based on synthetic oils with good lubricity and low reactivity with shales. Reservoir rocks are drilled with a fluid family known as drill-in, composed by saline polymeric solutions with bridging agents. One of the drilling fluid basic functions is to exert hydrostatic pressure over the permeable formations to avoid the formation fluid invasion to the well while the drilling operation takes place. The fluid pressure is normally kept above the formation pore pressure to prevent from kick events (formation fluid invasion to the well), that, in some cases, can lead to an uncontrolled influx (blowout). This concept, called overbalanced drilling, is traditionally employed in most of the drilling operations worldwide and in Brazil. As the bit penetrates the reservoir rock, the drilling fluid invades the formation due to the positive pressure differential between the well and the reservoir rock. Portions of the liquid phase of the drilling fluid are lost to the adjacent formation while part of the solids presented in drilling fluid, constituted by particles smaller than the formation pore size, penetrate the rock during the fluid loss period, rapidly plugging the region around the well (Martins, 2004). Larger particles accumulate on the wellbore walls, initiating an external cake formation. The filtrate and solid particles invasion during this process cause damage to formation around the well. The main purpose this paper is to describe a mathematical formulation which allows the estimation of filter cake permeability and non Newtonian invasion profile into the reservoir rock. The final goal is to couple experimental filter cake permeability data obtained from linear flow tests and radial flow modeling to optimize drilling fluid designed. 2. FUNDAMENTALS Consider a static filtration experiment, where a fluid, when submitted to constant differential pressure, flows through a porous medium previously saturated with the same fluid. The fluid volume that passes over the porous medium is monitored through the time and its rheological properties evaluated at the test temperature. Figure 2 shows the experimental scheme. Figure 2 shows the experimental scheme The fluid flow through to the porous media is normally describes by Darcy (1856): ρ µ k q f ef 1 + c q = −()∇p − ρ g (1) k µ f ef Where the k is the porous medium permeability, µef is the fluid viscosity, c is the Forchheimer (1901) constant and Re is the Reynolds number that can be describes by: ρ k q f Re = (2) µ ef and q is the superficial velocity that can be defined by: r q = vε (3) When the fluid flow is very slow i.e Re << 1 (4) Proceedings of COBEM 2005 20th International Congress of Mechanical Engineering Copyright © 2009 by ABCM November 15-20, 2009, Gramado, RS, Brazil Quadratic expression given by Eq. (1) can be approximated by: µ f q = −(∇p − ρ g) (5) k f 3. LINEAR UNIDIRECIONAL FLOW MODELING (LAB CONFIGURATION) During the reservoir rock drilling is normally used a drilling fluid type known as drill-in fluids. These fluids are composed by an aqueous base (normally brine) with calcium carbonate (adding different particle size distribution) and polymer additives that determine non-Newtonian behavior. Calcium carbonate and polymer additives help to form a low permeability filter cake and minimize drill in fluid invasion into reservoir. A filtrate sample was collected and taken as an example for the rheological characterization using a RS 600 Haake rheometer, using cone plate geometry. Figure 3 shows the flow curve and the power law approach to drill in fluid filtrate. 12 10 y = 0,3227x 0,5454 R2 = 0,9975 8 6 (Pa) τ τ τ τ 4 Drill in fluid 2 Power law model 0 0 100 200 300 400 500 γγγ (s -1 ) Figure 3 - Drill in fluid rheological behavior Figure 3 shows the power law approach with correlation coefficient (R 2) approximately equal to 1. Consistency index (M) and behavior index (n), respectively values equal to 0.3227 (Pa.s n) and 0.54, characterizing the non- Newtonian behavior of the filtrate sample evaluated. To obtain a model able of predicting the filter cake properties, monitoring the filtration parameters, were considered the following assumptions: • Filter cake thickness is defined by the solids concentration in the fluid and invasion volume. • Filter cake permeability is constant. • Hydrostatic effects are negligible. • Fluid filtrate presents Power Law behavior. • There is no solids invasion into the porous medium. Non Newtonian linear modeling : Integrating the Eq. (5) along the filter cake and porous medium, considering incompressible filter cake and neglecting the hydrostatic effects: h ( ) ∆ = µ ⋅ ⋅ pm + hcake t p ef q (6) kpm kcake Filtrate viscosity of a non Newtonian fluid can be estimated by a Power Law approach: µ = γ n−1 ef M. (7) The shear rate in the porous medium can be estimated, according to Massarani (1999), as a function of the superficial velocityr (q): γ = q (8) K pm Inserting the Eq. (8) in the Eq. (7): r n −1 q µ = M . (9) ef K pm Inserting the Eq. (9) in the Eq. (6): − r hpm h (t) ∆p = M.( q )n 1 q.. + cake (10) Kpm kpm kcake Rearranging the Eq. (10): ( ) q n hpm hcake t ∆ = + (11) p M.( .) Kpm . Kpm kpm kcake 1 n 1 1 1 () q n hpm h t ∆ n = n + cake (12) ( p) M .( ). K pm . K pm k pm kcake 1 n 1 1−n ∆ r hpm h ()t ( p ) n .( 1 ) n = q. + cake (13) M K pm k pm kcake Where: r dV = Q = 1 f = − dh cake q A A dt dt (14) Porosity is defined by: Vp ε = + (15) pm Vs Vp Porous volume can be obtained by: ε = pm .Vs V −ε (16) p 1( pm ) Filter cake volume can be defined by: = + A.hcake Vs Vp (17) Substituting the Eq. (16) in the Eq. (17): ε = + mp A.hcake Vs 1.( 1−εmp ) (18) or 1 A.h =V .( −ε ) (19) cake s 1 pm Differentiating Eq. (19) in relation to time: dh cake = dV s 1 A.